Endless self-avoiding walks

TL;DR

This paper introduces an endless self-avoiding walk model that eliminates end-effects, showing it shares key properties with traditional self-avoiding walks but with distinct growth characteristics, and suggests potential for broader applications.

Contribution

The paper presents a novel endless self-avoiding walk model with eliminated end-effects, providing new insights into its enumeration and scaling properties.

Findings

Same connective constant as standard self-avoiding walks

Critical exponent γ equals 1 exactly

Universal amplitude for number growth

Abstract

We introduce a self-avoiding walk model for which end-effects are completely eliminated. We enumerate the number of these walks for various lattices in dimensions two and three, and use these enumerations to study the properties of this model. We find that endless self-avoiding walks have the same connective constant as self-avoiding walks, and the same Flory exponent $\nu$. However, there is no power law correction to the exponential number growth for this new model, i.e. the critical exponent $\gamma = 1$ exactly. In addition, we have convincing numerical evidence to support the hypothesis that the amplitude for the number growth is a universal quantity, and somewhat weaker evidence which suggests that the number growth has no analytic corrections to scaling. The technique by which end-effects are eliminated may be generalised to other models of polymers such as interacting self-avoiding walks.